Bull. Astron. Belgrade
153 (l996), 29 – 34 UDC 524.882 Original scientific paper BOSE INSTABILITY IN KERR BLACK HOLES A. B. Gaina ERGO.,2 Deleanu str., ap.146, Chişinǎu 277050, Republic of Moldova phone (3732)639114, e-mail: acadm@mdearn.cri.md (Received: December 20, 1995) SUMMARY: Bose instability in rotating (Kerr) black holes (BH’s) consists in exponential increase in time of small perturbations of Bose mass fields, corresponding to superradiative, quasi bound levels. The minimal time of dumping of the angular momentum on the 2P envelope is much less than the time of dumping of the angular momentum by superradiation for primordial BH’s when the mass of M2 m << MP l . Very fast dumping of the angular momentum occures mM ≥ 0.203 (for π 0 ), 0.353 (η ), .0.065 (D0 ). Electrically charged when 0.46 ≥ M 2 the particles Pl particles cannot develop Bose instability due to the ionization of bound levels by electromagnetic radiation emitted by the BH itself. The neutral particles produce γ -bursts of energies 67.5, 274.5, 932 Mev correspondingly. The duration of bursts is 1.26 · 10−17 s(π 0 ), 2.99 · 10−18 s(η ), 8.55 · 10−19 s(D 0 ). The radiated energies are 1.20 · 1035 erg, 8.67 · 1034 erg, erg 8.55 · 1033 erg, corresponding to powers of the order of magnitude 1052 s . Other consequences for BH’s evaporation are discussed. 1. INTRODUCTION Bose instability of rotating Black Holes (Kerr BH’s) is related with an exponential increase in time of small perturbations of a test mass field, corresponding to superradiative quasibound states with energies: E ≤ min{µc2 , h̄mΩH } (1) where µ and E are respectively the rest mass and energy of the particles, m is the projection of the momentum on the BH’s axis, M and J = M ac are mass and angular momentum of the BH and the angular velocity of the BH is written as ΩH = ac3 , 2GM r+ (2) where 1 r+ = GM/c2 + (G2 M 2 /c4 − a2 ) 2 . On the level of a second quantized quantum field theory (in brief QFT) this corresponds to the occurrence of spontaneous and induced particles creation proce29 A. B. GAINA sses on quasi bound superradiative levels. Only spontaneous generation of fermions may occur due to Dirac exclusion principle, but bosons may accumulate on such levels by induction (or stimulation). On the level of Klein-Gordon,Dirac and other similar QF equations this corresponds to the fact that s = 21 , 32 , ... mass equations support only dumping (Gaina et al. 1980) while the s = 0, 1, 2, .. mass equation may change the sign of the imaginary part of the energy (Ternov et al. 1978): (0) E = Enlm − iγnlm , (3) E (0) ≡ ReE < µc2 , γ ≡ ImE = 1 3 , , ...; 2 2 > h̄mΩH ;      > 0, for s = > 0, for s = 0, 1,...and E (0)     ≤ 0, for s = 0, 1,...and E (0) ≤ h̄mΩH . In other words, bosons supports self-stimulated generation (and in consequence -accumulation) on quasi-bound superradiative states (1) in which the wave function increase as Φ ∼ eλt , where λ = −γ, for E 0 ≤ h̄mΩH and the number of particles and the energy density increase as:  √ 1 N = i {Φ∗ (∂ 0 Φ) − Φ(∂ 0 Φ)∗ } −gd3 x ∼ e2λt C, 2 (4) E=  √ ν k(t) T·ν0 −gd3 x =  √ T·00 −gd3 x ∼ e2λt , (5) ν where k(t) = δtν is time like Killing vector of the Kerr metrics. One could show by an alternative way that the probability of the transition of a system (BH + bosons) from an initial state |N− → , 0 > with N− → k →k − bosons with quantum number k and 0 antibosons into a final state |N− → + 1, 1 > with N− → + 1 bosons k k and 1 antiboson will be proportional to the square of the matrix element   < 0, nk |T·00 |nk + 1, 1 >2 = |c|2 (Nk + 1). (6) When N− → = 0, this is just the probability of a sponk taneous generation of a pair boson-antiboson from which one particle localized on quasibound state and other inside the BH. Otherwise (6) gives the probability of self-stimulated generation of pairs. So, one could take into consideration the following equations for the number of particles on the superradiative levels, mass, angular momentum variations of a BH (Gaina, 1989): dNnlm = λnlm (Nnlm + 1), dt 30 (7)  d(M c2 ) (0) =− λnlm Enlm (Nnlm + 1), dt (8)  dJ =− λnlm h̄m(Nnlm + 1). dt (9) nlm nlm Equation (7) gives the number of particles on the quasilevel with quantum numbers n ≡ 1 + l + nr , l, m (nr = 0, 1, 2, ...; l = 0, 1, 2, ...) for scalar bosons. Generally, equations (7)-(9) are nonlinear, admitting solutions only in very special cases. 2. BOSE INSTABILITY IN KERR BH’S Let us restrict ourselves here to examination only of scalar bosons, as the solutions for the vector bosons and other boson mass fields are still unknown in Kerr backgrounds. As it was shown (Gaina, 1989) only the case µM  MP2 l ≡ h̄c G (10) is of interest, if one excludes the case of very large µ → MP l . The probabilities of particles generation were calculated in Gaina et al. (1980) (see also Detweiler, 1980; Gaina and Kochorbe, 1987). The main contribution to the change of mass and angular momentum of the BH gives rise to the generation and accumulation of particles on the 2P level. The dynamic equations for the number of particles and angular momentum are (we use below the system of units c = h̄ = G = 1 ): dNnp = λnp (Nnp + 1) ; dt (11) dN2p dJ ≃− (12) dt dt while the mass change is negligible. By using the law of conservation of total an Nnp in the system gular momentum J = J0 − n BH+bosons we obtain the law of variation of the number of particles and angular momentum of the BH in explicit form: Nnp = J0′ where 1 − exp − (J0′ + 1) µ9 M 6 t/48 1 + J0′ exp [− (J0′ + 1) µ9 M 6 t/48] (13) J = J0 − N2p (14) Jo′ = J0 − Jst , (15) BOSE INSTABILITY IN KERR BLACK HOLES Jst being the BH angular momentum at which superradiance at the given level stops. The exact value of Jst is : Jst = 4E (0) M 3 m , m2 + 4(E (0) )2 M 2 (16) 0 while for the case µM << 1 one obtains: Jst ≃ 4µM 3 << M 2 . (17) Note that the time of dumping of the angular momentum of the BH into the levels is τJ ≈ 48tP l MP l µ 3 MP2 l µM 6 ln (J0 − Jst + 1) ; (J0 − Jst ) (18) which approximately equals τJ ≈ 96tP l MP2 l µM MP l µ 8 ln M MP l (19) when µM << MP2 l . The mass of the envelope of bosons on 2P state is △M = M0 − Mst ≈ µN2p ≈ µJ0′ ≈ µ(J0 − Jst ) ≈ µM a0 . It will be much less than the mass of the BH itself if µM << MP2 l . The discussion of other details of the dumping of the angular momentum of the BH, caused by Bose instability is given in Gaina (1989). The time of loss of the angular momentum of a rotating black hole by superrradiation (see Zel’dovitch, 1971) is τsuperrad ∼ 8πe M MP l ξ 3 tP l , (20) where ξ is of order unity. Then, the ratio M MP l 2 Mµ MP2 l 9 M MP l (21) may be much greater than unity if M ≫ MP l . We do not now consider the cases µ  MP l , M MP l and µM ∼ MP2 l . For the latest one we can give some estimations based on analytical approaches developed in Gaina et al. (1980), Ternov et al. (1978), Gaina (1989) and Zouros and Eardley (1979), while an exact treatment should be given numerically. The probability of pair production for the case µM ≫ MP2 l was calculated by Zouros and Eardley (1979) for scalar bosons and improved by Gaina (1989) . Tunneling probability near the threshold of pair production for E  µ, a → M (but a = M ), and l − m ≈ |m − m0 | << m0 = µ/ΩH has the form (Gaina, 1989). τsuperrad π ξ = e τJ 12 ln −1 sign(m − m0 ) λm (a) = 10−7 M    (22)  √ m − m0    exp{−2πµM 21 − − 2 }.  m  a2 1− M 2 Here we omit the weaker dependence on the orbital quantum number and the particle energy. As m → m0 the exponent in eq. (5) changes into the result in Zouros and Eardley (1979) to within a factor of two in the exponent. The corresponding time of relaxation (dumping) of angular momentum for an extremely rapidly rotating black hole with unfilled levels is less than the age of the universe for BHs with masses µM  (23 ÷ 26)MP2 l . The characteristic range of variation of the specific angular momentum of the BH is 0.6 a/M < 1. One should emphasize that thermal effects will be small  if the temperature of BH: kTBH = 1 − a2 /M 2 /4πrt << E (0) ≈ µc2 . From this it is easy to obtain the criterion for macroscopic tunneling:  2 1 − (ac2 /GM ) << 4πµM/MP2 l . (23)  rapidly rotating It 2can be satisfied easily for ac → GM or macroscopic µM > MP2 l back holes. 3. THE ENERGETIC SPECTRUM OF QUASI BOUND LEVELS There are very  the  different energy spectra in / long wave length µM << MP2 l , or r+ << λc and   / short wave length µM ≫ MP2 l , or r+ ≫ λc limits. In the first case we have a full hydrogenlike spectrum for a = 0 µ2 M 2 En0 . (24) =1− µ 2n2 S quasibound levels appear for µM  0.25MP2 l , P quasibound levels appear for µM  0, 46MP2 l (see Zouros and Eardley, 1979), D quasibound levels ap2 pear for µM  0, 74M P l and so on, nl quasilevel √ appear for µM  63 MP2 l if l ≫ 1 (quasiclassical limit). Such a criterion for the Kerr metrics it is still unknown. The extremely rotating Kerr BH were examined in Gaina and Zaslavskii (1992). It was shown that the marginally stable corotating orbit is dumped for Klein-Gordon particles. 31 A. B. GAINA However, it is known (Gaina, 1989a) that the spectrum (24) is a good approximation also for Kerr metric if 1 µM << l + . (25) 2 So, one could expect that the criterion for the existence of a 2P and 3D quasibound levels is roughly the same as for a Shwarzshild BH. 4. ELEMENTARY PARTICLES AND THE MASSES RANGES FOR BOSE INSTABILITY Assuming µM ≈ 0.45 one derives from eq. (18) the minimal dumping time of the angular momentum for an extremely rotating BH: τJ(min) = 5.7 · 104 µ−1 ln M MP l . (26) By taking into account the time life of the most of mesons τL < 10−8 one obtains that boson instability cannot develop for BHs with masses M > 5MP2 l /µπ = 3.8 · 1016 g. If we assume, however, that boson instability occurs for neutrino pairs (assuming the neutrino to have mass) with a characteristic time determined by equation (26), we obtain an upper limit on the mass of a BH subjected to boson instability: M < 2.4 · 1023 g. For known bosons the masses ranges for black holes subjected to Bose instability are given in Table 1. It is easy to see that the bosonic instability cannot develop for η -meson, W ± and Z 0 due to very short lifetime. On the other hand one should give a rigorous treatment of η decay near the horizon of the black hole which could warrant our expectations for accumulations of η mesons. One must take into account also that estimations for the upper limit mass of the BH subjected to vector instability were made on the basis of scalar equation and the actual value of the τJ may be less for W ± and Z 0 . Table 1. Masses Ranges for the Known Bosons Particle π± π0 η 0 K 0, K 0 D D± F± W± Z0 Life time τL (s) Low limit mass (g) Upper limit mass (g) 2.6 · 10−8 0.8 · 10−16 2.4 · 10−19 10−9 5 · 10−13 10−12 2 · 10−13 3 · 10−25 −//− 5.8 · 1013 7.0 · 1014 3.0 · 1014 2.1 · 1013 1.6 · 1013 1.5 · 1013 9.4 · 1012 4.8 · 1012 4.2 · 1012 8.5 · 1014 8.85 · 1014 2.18 · 1014 2.4 · 1014 6.42 · 1013 6.4 · 1013 5.9 · 1013 1.2 · 1012 1.08 · 1012 Mass of the particles (Mev) 5. STOPPING THE INSTABILITY FOR ELECTRICALLY CHARGED PARTICLES. ELECTROMAGNETIC TRANSITIONS AND PHOTOIONIZATION Of course, we do not take into account the annihilation of π ± , K ± , D± and F ± during their generation and accumulation near the Black Hole. Particles of opposite charges may annihilate rapidly during their generation. But one must take into account the influence of a strong gravitational field near the horizon of a black hole. Let us assume that electrically charged particles and their antiparticles could accumulate on quasilevels, for instance, on 2P quasibound level in 32 140 135 549 498 1864 1869 2020 83 · 103 93 · 103 the field of an highly rotating BH (a → M ) of mass M  0.45MP2 l /µ , i.e. near the upper limit of Bose instability. In this case one has 3 processes which can stop the instability: i) electromagnetic transitions 2p → 1s, or other transitions on nonsuperradiative levels; ii) annihilation of particles into two fotons (π + + π − → 2γ, K + + K − → 2γ, and so ones) ; iii) photoionization of bound levels by the electromagnetic radiation emitted by the BH itself. The equation which governs the number of particles on 2P level is: dN2p =λ2p (N2p + 1) − W2p→1s N2p − dt 2 Wann N2p − Wion N2p . (27) BOSE INSTABILITY IN KERR BLACK HOLES It is not difficult to calculate the probability of normal dipole transitions 2p → 1s in the field of a such black hole assuming, that a hydrogenlike spectrum of bound states is realized. One could note that there are also anomalous transitions with △m = 0 which may have the same order of magnitude in the field of an extremely rotating BH (Chizov and Myshenkov, 1991) but we do not examine here such transitions. The probability W2p→1s will be: 2 3 W2p→1s = αµ(µM )4 . (28) Transitions (28) impose further restraints on the masses of BH’s supposed to Bose instability:  M2 210 α MP2 l 4 M≥ = 0.24 P l , (29) 7 3 µ µ that is the actual low limit mass for instability will be greater for π ± , D± , F ± , K ± . The total cross section of annihilation of particles of opposite charges on the quasilevels may be also very easily estimated in the nonrelativistic limit: σann = πα2 α2 ≈ . 2µ2 2µE (0) Akhiezer and Berestetsky (1981) give an exact formula for the cross section of ionization of 2P atomic levels. In the case of N 2P electrons one has: σ2p =N2p 210 π 2 α 9µI2p Irp h̄ω 5 3+8 Irp h̄ω η e−4ηarcctg 2 . 1 − exp(−2πη) (30) 28 απ 3µIrp Irp ω σ2p = √ 12 2  µ 9/2 ω N2p . (µM )7 N2p . 8M 3 ω 3 dnp = . (34) dtdw 9π After the integration of (32) with S = a20 = µ4 1M 2 one has: 8α 11 µ (µM ) . (35) 81π For N2p particles localized on 2P level this formula must be multiplied by N2p. Thus, photoionization is slow compared with dipole transitions for small occupation numbers, but may suppress the last ones for great N2p . It is easy to estimate the number of particles on the 2P level after the photoionization ignoring the dipole transitions and annihilation of pairs. The equation (29) will have the form: (ion) W2p = dN2p (ion) 2 = λ2p (N2p + 1) − W2p N2p . (36) dt Assuming N2p ≫ 1 one obtains for the equilibrium number of particles: Nrp = λ2p (ion) Wrp = 81π −3 (µM ) , 96α (37) N2p = 3985. 9/2 Adapting this formula for a BH we obtain the cross section of photoionization of one particle on 2P  level Zα → µM/MP2 l : / πα(λc )2 1 Γ1ω1mp dnp = < n >= = dtdw 2π 2π (33) 4 8πM r+ 2 3 = M (ω − Ω ) ω H 9 2π 2 is the number of p-photons emitted by the BH (Akhiezer and Berestetsky, 1981). In the limit a → M one has: that is for a BH with incipient instability (µM ≈ 0, 45) one derives In the limit h̄ω ≫ Irp one has approximately: σ2p ≃ where (31) An important feature of the cross section (31) is the dependence of the rate of ionization on the number of particles on the level. Let us calculate now the probability of ionization of one electrically charged 2P scalar particle by electromagnetic radiation emitted by the BH itself as a result of superradiance. The probability will be:  1 dnp (ion) σ2p dw Wrp (32) = S dtdw So, photoionization stops efficiently Bose instability for electrically charged mesons π ± , K ± , D± , F ± . 6. CONCLUSIONS From the Table 1 it is easy to see that the bosonic instability cannot develop for η -meson, W ± and Z 0 due to very short lifetime. On the other hand one should give a rigorous treatment of η decay near the horizon of the black hole which could warrant our expectations for accumulations of η mesons. One must take into account also that estimations for the upper limit mass of the BH subjected to vector instability were made on the basis of scalar equation and the actual value of the τJ may be less for W ± and Z 0 . 33 A. B. GAINA Self stimulated generation and accumulation of bosons in the field of highly rotating black holes (Bose instability) is an efficient mechanism of dropping of angular momentum for primordial black holes. Electrically charged particles cannot accumulate near the black hole due to electromagnetic transitions, annihilations and photoionization. But π 0 , D0 , K 0 could rapidly accumulate near black holes and produce after γ bursts of powers: dE dt int ≃ 1 c5 288 G µM MP2 l 10 1 , ln MMP l that is ≃ 9, 5 · 1051 erg/s (for π 0 ), 2.9·1052 egr/s (for η), 1052 erg/s (for D0 ), if one assume BH to be near the threeshold of instability (i.e. µM ≈ 0.45MP2 l). This corresponds to effective masses of radiated energies: 133M t (for D0 ). The energies of γ-photons radiated will be 67,5 M ev (for π 0 ), 274, 5M ev (for η), 932M ev (for D0 ). REFERENCES Akhiezer, A. I. and Berestetsky, V. B.: 1981, Quantum Electrodynamimics, Nauka, Moscow. Chizov, G. A., Myshenkov, O.: 1991, Preprint of the Physics Dept. Moscow State University. Detweiler, S.: 1980, Phis. Rev., D 22, 2323. Gaina, A.: 1989, Pis’ma Astron. Zh. 15, 567 [Engl. transl.: Sov. Astron. Lett. 15(3), 567 (1989)]. Gaina, A: 1989a, Quantum Particles in Einstein-Maxwell Fields Stiinta, Chisinau. Gaina, A. and Kochorbe, F. G.: 1987, Zh. Exp. Teor. Fiz. 92, 369. Gaina, A., Ternov, I. M. and Chizhov, G. A.: 1980, Izv. Vyssh. Uchebn. Zaved., Fiz. 23, No. 8, 56 [Engl. transl.: Sov. Phys. J. 23, No 8 (1980)]. Gaina, A. and Zaslavskii, O. B.: 1992, Class. Quantum. Grav. 9, 667. Starobinsky, A. A. and Churilov, S. M.: 1973, Zh. Exp. Teor. Fiz. 65, 3. Ternov, I. M., Khalilov, V. R., Chizhov, G. A. and Gaina, A. B.: 1978, Izv. Vyssh. Uchebn. Zaved., Fiz. 21, No. 9, 109 [Engl. transl.: Sov. Phys. J. 21, No. 9 (1980)]. Zel’dovitch, Ya. B.: 1971, Pis’ma Zh. Exp. Teor. Fiz. 14, 270 (1971); Zh. Exp. Teor. Fiz. 62, 2076 (1972). Zouros, T. and Eardley, D. M.: 1979, Ann. Phys. (N.Y.) 118, 139. BOZEOVA NESTABILNOST U KEROVIM CRNIM RUPAMA A. B. Gaina ERGO.,2 Deleanu str., ap.146, Chişinǎu 277050, Republic of Moldova UDK 524.882 Originalni nauqni rad Bozeova nestabilnost u rotiraju
im (Kerovim) crnim rupama (CR) sastoji se u eksponencijalnom porastu u vremenu, malih perturbacija poƩa Bozeove mase, koja odgovaraju superradijativnim, kvazivezanim nivoima. Minimalno vreme priguxeƬa ugaonog momenta 2P Ʃuske je mnogo maƬe nego vreme priguxeƬa ugaonog momenta superradijacijom, za primordijalne CR kada je masa qestica M2 m << MP l . Veoma brzo priguxeƬe ugaonog momM ≥ 0.203 (za menta se dexava kada je 0.46 ≥ M 2 Pl 34 π 0 ), 0.353 (η), .0.065 (D0 ). Naelektrisana qestica ne moжe da razvije Bozeovu nestabilnost usled jonizacije vezanih nivoa elektromagnetnim zraqeƬem koje emituje sama CR. Neutralne qestice stvaraju γ–bƩeskove sa energijama 67.5, 274.5, 932 Mev respektivno. TrajaƬe bƩeskova je 1.26 · 10−17 s (π 0 ), 2.99 · 10−18 s (η), 8.55 · 10−19 s (D0 ). Izraqene energije su 1.20 · 1035 erg, 8.67 · 1034 erg, 8.55 · 1033 erg, xto odgovara snagama reda veliqine 1052 erg s . Diskutovane su i druge posledice isparavaƬa crnih rupa.